Question

the graph of a polynomial function f is shown. select all the true statements about the polynomial. the degree of the polynomial is even. the degree of the polynomial is odd. the leading coefficient is positive. the leading coefficient is negative. the constant term of the polynomial is positive. the constant term of the polynomial is negative.

Explanation:

Step1: Analyze end - behavior for degree

As \(x\to\pm\infty\), the graph of the polynomial falls to the left and falls to the right. For a polynomial \(y = a_nx^n+\cdots+a_0\), if \(n\) is even and \(a_n<0\), the end - behavior is \(y\to-\infty\) as \(x\to\pm\infty\). So the degree \(n\) is even.

Step2: Determine leading coefficient sign

Since the graph falls to the left and falls to the right (\(y\to-\infty\) as \(x\to\pm\infty\)), the leading coefficient \(a_n\) is negative.

Step3: Find the constant term

The constant term of a polynomial \(y = a_nx^n+\cdots+a_0\) is the \(y\) - intercept. The graph intersects the \(y\) - axis above the origin (\(y\) - intercept is at \(y = 3\)), so the constant term is positive.

Answer:

The degree of the polynomial is even.
The leading coefficient is negative.
The constant term of the polynomial is positive.